Chapter 10, Problem 32QP

Summary Introduction

To find: The net present value and the internal rate of return.

Introduction:

The variation between the present value of the cash outflows and the present value of the cash inflows are known as the net present value. In capital budgeting the net present value is utilized to analyze the profitability of a project or investment. The rate of return which equates the initial investment and the present value of net cash inflows are referred to as internal rate of return. This is also called as actual rate of return.

Answer to Problem 32QP

The net present value is $4,725,575.74 and the internal rate of return is 27.52%.

Explanation of Solution

Given information:

Company A projects the unit sale for the new 7 octave voice emulation implant as follows:

• The year 1 unit sales is 84,000
• The year 2 unit sales is 98,000
• The year 3 unit sales is 113,000
• The year 4 unit sales is 106,000
• The year 5 unit sales is 79,000

The production implant needs $1,500,000 in the net working capital to begin their production activities. The extra net working capital investment for every year is equivalent to the 15% of the sales that is projected has to rise for the following year. The total fixed cost is $3,400,000 for a year, the unit price is $395, and the variable production cost is $265. The installation cost of the equipment is $17,000,000.

The equipment is qualified in the 7 Year MACRS depreciation under the property class. In 5 years the equipment sale can be sold for 20% of its acquisition cost. The marginal tax bracket is 35 percent and has the required rate of return which is 18%.

MACRS depreciation table for year 7:

MACRS Depreciation table for seven year
Year	Seven year
1	14.29%
2	24.49%
3	17.49%
4	12.49%
5	8.93%
6	8.92%
7	8.93%
8	4.46%

Computation of the net present value:

Computation of the cash outflow:

$\begin{array}{c}\text{Cash outflow = }\left(\begin{array}{l}-\text{Capital spending}\left(\text{Installed cost}\right)+\\ -\text{Initial net working capital}\end{array}\right)\\ =\left(-\$17,000,000\right)+\left(-\$1,500,000\right)\\ =-\$18,500,000\end{array}$

Table showing the cash inflows:

Year	1	2	3	4	5
Ending book value	$14,570,700	$10,407,400	$7,434,100	$5,310,800	$3,792,700
Sales	$33,180,000	$38,710,000	$44,635,000	$41,870,000	$31,205,000
Less: Variable costs	-$22,260,000	-$25,970,000	-$29,945,000	-$28,090,000	-$20,935,000
Fixed costs	-$3,400,000	-$3,400,000	-$3,400,000	-$3,400,000	-$3,400,000
Depreciation	-$2,429,300	-$4,163,300	-$2,973,300	-$2,123,300	-$1,518,100
EBIT	$5,090,700	$5,176,700	$8,316,700	$8,256,700	$5,351,900
Less: Taxes	-$1,781,745	-$1,811,845	-$2,910,845	-$2,889,845	-$1,873,165
Net income	$3,308,955	$3,364,855	$5,405,855	$5,366,855	$3,478,735
Add: Depreciation	$2,429,300	$4,163,300	$2,973,300	$2,123,300	$1,518,100
Operating cash flow	$5,738,255	$7,528,155	$8,379,155	$7,490,155	$4,996,835
Net cash inflows:
Operating cash flow	$5,738,255	$7,528,155	$8,379,155	$7,490,155	$4,996,835
Change in net working capital	–829,500	–888,750	$414,750	$1,599,750	$1,203,750
Capital spending	$0	$0	$0	$0	$3,537,445
Total cash inflows	$4,908,755	$6,639,405	$8,793,905	$9,089,905	$9,738,030

Computations for the above table:

Formula to calculate the ending book value:

$\text{Ending book value = }\left[\begin{array}{l}\text{ Installed cost of an equipment}-\\ \left(\text{Installed cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]$

Computation of the ending book value for year 1:

$\begin{array}{c}\text{Ending book value for year 3 = }\left[\begin{array}{l}\text{ Ending book value of an equipment for year 2}-\\ \left(\text{Initial cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$10,407,400}-\left(\text{\$17,000,000}\times 17.\text{49\%}\right)\\ \text{=\$10,407,400}-\$2,973,300\\ =\$7,434,100\end{array}$

$\begin{array}{c}\text{Ending book value for year 1 = }\left[\begin{array}{l}\text{ Installed cost of an equipment}-\\ \left(\text{Installed cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$17,000,000}-\left(\text{\$17,000,000}\times \text{14}\text{.29\%}\right)\\ \text{=\$17,000,000}-\$2,429,300\\ =\$14,570,700\end{array}$

$\begin{array}{c}\text{Ending book value for year 2 = }\left[\begin{array}{l}\text{ Ending book value of an equipment for year 1}-\\ \left(\text{Initial cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$14,570,700}-\left(\text{\$17,000,000}\times 2\text{4}\text{.49\%}\right)\\ \text{=\$14,570,700}-\$4,163,300\\ =\$10,407,400\end{array}$

$\begin{array}{c}\text{Ending book value for year 4 = }\left[\begin{array}{l}\text{ Ending book value of an equipment for year 3}-\\ \left(\text{Initial cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$7,434,100}-\left(\text{\$17,000,000}\times 12.\text{49\%}\right)\\ \text{=\$7,434,100}-\$2,123,300\\ =\$5,310,800\end{array}$

$\begin{array}{c}\text{Ending book value for year 5 = }\left[\begin{array}{l}\text{ Ending book value of an equipment for year 4}-\\ \left(\text{Initial cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$5,310,800}-\left(\text{\$17,000,000}\times 8.93\text{\%}\right)\\ \text{=\$5,310,800}-\$1,518,100\\ =\$3,792,700\end{array}$

Formula to calculate the sales:

$\text{Sales = Unit sales}\times \text{Price per unit}$

Computation of the sales:

$\begin{array}{c}\text{Sales for year 1 = Unit sales for year 1}\times \text{Price per unit}\\ \text{=\$84,000}\times \$395\\ =\$33,180,000\end{array}$

$\begin{array}{c}\text{Sales for year 2 = Unit sales for year 2}\times \text{Price per unit}\\ \text{=\$98,000}\times \$395\\ =\$38,710,000\end{array}$

$\begin{array}{c}\text{Sales for year 3 = Unit sales for year 3}\times \text{Price per unit}\\ \text{=\$113,000}\times \$395\\ =\$44,635,000\end{array}$

$\begin{array}{c}\text{Sales for year 4 = Unit sales for year 4}\times \text{Price per unit}\\ \text{=\$106,000}\times \$395\\ =\$41,870,000\end{array}$

$\begin{array}{c}\text{Sales for year 5 = Unit sales for year 5}\times \text{Price per unit}\\ \text{=\$79,000}\times \$395\\ =\$31,205,000\end{array}$

Computation of the depreciation:

The depreciation amount is calculated by using the MACRS depreciation table for seven years.

Formula to calculate depreciation:

$\text{Depreciaiton = }\left(\begin{array}{l}\text{Installation cost of an equipment}\\ \times \text{MACRS rate for the specific year}\end{array}\right)$

Computation of the depreciation:

$\begin{array}{c}\text{Depreciaiton for year 1 = }\left(\begin{array}{l}\text{Installation cost of an equipment}\\ \times \text{MACRS rate for year 1}\end{array}\right)\\ \text{=\$17,000,000}\times 14.29\%\\ =\$2,429,300\end{array}$

$\begin{array}{c}\text{Depreciaiton for year 2 = }\left(\begin{array}{l}\text{Installation cost of an equipment}\\ \times \text{MACRS rate for year 2}\end{array}\right)\\ \text{=\$17,000,000}\times 24.29\%\\ =\$4,163,300\end{array}$

$\begin{array}{c}\text{Depreciaiton for year 3 = }\left(\begin{array}{l}\text{Installation cost of an equipment}\\ \times \text{MACRS rate for year 3}\end{array}\right)\\ \text{=\$17,000,000}\times 17.49\%\\ =\$2,973,300\end{array}$

$\begin{array}{c}\text{Depreciaiton for year 4 = }\left(\begin{array}{l}\text{Installation cost of an equipment}\\ \times \text{MACRS rate for year 4}\end{array}\right)\\ \text{=\$17,000,000}\times 12.49\%\\ =\$2,123,300\end{array}$

$\begin{array}{c}\text{Depreciaiton for year 5 = }\left(\begin{array}{l}\text{Installation cost of an equipment}\\ \times \text{MACRS rate for year 5}\end{array}\right)\\ \text{=\$17,000,000}\times 8.93\%\\ =\$1,518,100\end{array}$

Formula to calculate the taxes:

$\text{Taxes = EBIT for the specific year}\times \text{Marginal tax rate}$

Computation of the taxes:

$\begin{array}{c}\text{Taxes for year 1 = EBIT for year 1}\times \text{Marginal tax rate}\\ \text{=\$5,090,700}\times \text{35\%}\\ \text{=\$1,781,745}\end{array}$

$\begin{array}{c}\text{Taxes for year 2 = EBIT for year 2}\times \text{Marginal tax rate}\\ \text{=\$5,176,700}\times \text{35\%}\\ \text{=\$1,811,845}\end{array}$

$\begin{array}{c}\text{Taxes for year 3 = EBIT for year 3}\times \text{Marginal tax rate}\\ \text{=\$8,316,700}\times \text{35\%}\\ \text{=\$2,910,845}\end{array}$

$\begin{array}{c}\text{Taxes for year 4 = EBIT for year 4}\times \text{Marginal tax rate}\\ \text{=\$8,256,700}\times \text{35\%}\\ \text{=\$2,889,845}\end{array}$

$\begin{array}{c}\text{Taxes for year 5 = EBIT for year 5}\times \text{Marginal tax rate}\\ \text{=\$5,351,900}\times \text{35\%}\\ \text{=\$1,873,165}\end{array}$

Formula to calculate the net working capital:

$\text{Net working capital = Increases of 15\%}\left(\text{Sales for the next year}-\text{Sales for the current year}\right)$

Computation of the net working capital:

$\begin{array}{c}\text{NWC for year 1}=\text{ Increases of 15\%}\left(\text{Sales for year 2}-\text{Sales for year 1}\right)\\ =.15\times \left(\$38,710,000-\$33,180,000\right)\\ =.15\times \$5,530,000\\ =\$829,500\end{array}$

$\begin{array}{c}\text{NWC for year 2}=\text{ Increases of 15\%}\left(\text{Sales for year 3}-\text{Sales for year 2}\right)\\ =.15\times \left(\$44,635,000-\$38,710,000\right)\\ =.15\times \$5,925,000\\ =\$888,750\end{array}$

$\begin{array}{c}\text{NWC for year 3}=\text{ Increases of 15\%}\left(\text{Sales for year 4}-\text{Sales for year 3}\right)\\ =.15\times \left(\$41,870,000-\$44,645,000\right)\\ =.15\times \$2,765,000\\ =\$414,750\end{array}$

$\begin{array}{c}\text{NWC for year 4}=\text{ Increases of 15\%}\left(\text{Sales for year 5}-\text{Sales for year 4}\right)\\ =.15\times \left(\$31,205,000-\$41,870,000\right)\\ =.15\times \$10,665,000\\ =\$1,599,750\end{array}$

Computation of the net working capital for the year 5:

$\begin{array}{c}\text{NWC for year 5 = Recovery of all NWC}\\ \text{= Initial NWC – }\left[\begin{array}{l}\text{Change in NWC for year 1 }\\ \text{+ Change in NWC for Year 2 }\\ \text{+}\text{Change in NWC for year 3 }\\ \text{+ Change in NWC for Year 4}\end{array}\right]\end{array}$

$\begin{array}{c}=\$1,500,000-\left[\begin{array}{l}-\$829,500-\$888,750\\ +414,750+\$1,599,750\end{array}\right]\\ =\$1,500,000-\$296,250\\ =\$1,203,750\end{array}$

Computation of the ending book value:

$\begin{array}{c}\text{Ending book value = }\left[\begin{array}{l}\text{Installation cost of an equipment}-\\ \left(\begin{array}{l}\text{Depreciation for year 1}+\text{Depreciation for year 2+}\\ \text{Depreciation for year}3+\text{Depreciation for year}4+\\ \text{Depreciation for year}5\end{array}\right)\end{array}\right]\\ =\$17,000,000\left(\begin{array}{l}\$2,429,300+\$4,163,300+\\ \$2,973,300+\$2,123,300+\$1,518,100\end{array}\right)\\ =\$17,000,000-\$13,207,300\\ =\$3,792,700\end{array}$

Formula to calculate the after-tax salvage value:

$\text{After-taxsalvage value = }\left[\begin{array}{l}\text{Total fixed cost +}\left(\begin{array}{l}\text{Ending book value}-\\ \text{Total fixed cost}\end{array}\right)\\ \times \text{Marginal tax rate}\end{array}\right]$

Computation of the after-tax salvage value:

$\begin{array}{c}\text{After-taxsalvage value = }\left[\begin{array}{l}\text{Total fixed cost +}\left(\begin{array}{l}\text{Ending book value}-\\ \text{Total fixed cost}\end{array}\right)\\ \times \text{Marginal tax rate}\end{array}\right]\\ =\$3,400,000+\left(\$3,792,700-\$3,400,000\right)\times .35\\ =\$3,400,000+\$137,445\\ =\$3,537,445\end{array}$

Formula to calculate the net present value:

$\begin{array}{c}\text{NPV = Cash outflow +<x-custom-btb-me data-me-id="2578" class="microExplainerHighlight">Cash inflow</x-custom-btb-me>}\\ \text{=}-\text{Total cash outflow +}\left(\begin{array}{l}\frac{\text{Cash inflow for year 1}}{{\left(1+\text{r}\right)}^1}\text{+}\\ \frac{\text{Cash inflow for year 2}}{{\left(1+\text{r}\right)}^2}\text{+}\\ \frac{\text{Cash inflow for year 3}}{{\left(1+\text{r}\right)}^3}\text{+}\\ \frac{\text{Cash inflow for year 4}}{{\left(1+\text{r}\right)}^4}\text{+}\\ \frac{\text{Cash inflow for year 5}}{{\left(1+\text{r}\right)}^5}\end{array}\right)\end{array}$

Computation of the net present value:

$\begin{array}{c}\text{NPV = Cash outflow +Cash inflow}\\ \text{=}-\text{Total cash outflow +}\left(\begin{array}{l}\frac{\text{Cash inflow for year 1}}{{\left(1+\text{r}\right)}^1}\text{+}\\ \frac{\text{Cash inflow for year 2}}{{\left(1+\text{r}\right)}^2}\text{+}\\ \frac{\text{Cash inflow for year 3}}{{\left(1+\text{r}\right)}^3}\text{+}\\ \frac{\text{Cash inflow for year 4}}{{\left(1+\text{r}\right)}^4}\text{+}\\ \frac{\text{Cash inflow for year 5}}{{\left(1+\text{r}\right)}^5}\end{array}\right)\\ =-\$18,500,000+\left(\begin{array}{l}\frac{\$4,908,755}{{\left(1.18\right)}^1}+\frac{\$6,639,405}{{\left(1.18\right)}^2}+\\ \frac{\$8,793,905}{{\left(1.18\right)}^3}+\frac{\$9,089,905}{{\left(1.18\right)}^4}+\frac{\$9,738,030}{{\left(1.18\right)}^5}\end{array}\right)\\ =\$18,500,000+\left(\begin{array}{l}\$4,159,961.86+\$4,768,317.29+\\ \$5,352,242.07+\$4,688,471.86+\$4,256,582.66\end{array}\right)\\ \end{array}$

$\begin{array}{c}=\$18,500,000+\$23,225,575.74\\ =\$4,725,575.74\end{array}$

Hence, the net present value is $4,725,575.74.

Computation of the internal rate of return:

The internal rate of return is calculated by the spreadsheet method.

Step 1:

• Type the formulae of the internal rate of return in H6 in the spreadsheet and consider the IRR value as H8

Step 2:

• Assume the IRR value as 0.10%

Step 3:

• In the spreadsheet go to data and select the what-if analysis.
• In what-if analysis select goal seek
• In set cell select H6 (the formulae)
• The To value is considered as 0
• The H8 cell is selected for the by changing cell.

Step 4:

• Following the previous step click OK in the goal seek. The goal seek status appears

Step 5:

• The IRR value appears to be 27.5164749401034%

Hence, the internal rate of return is 27.52%.